Stochastic difference equation?

This is a question about one single step of a solution of a long equation.

http://www.geocities.com/link_herooftime/math.jpg

where P, U and V are variables. a, b, c, d are constants and t is the time, which are measured in discrete periods.

The question is how to go from equation 1 to equation 3, and how the L appears.

Attempted solution

I have solved it until equation 2, and I see that the solution requires the extraction of P out of the bracket on the left hand side. Problem becomes how to do it, since the P minus one period doesnt equal P of current period. Nevertheless the key somehow does this. Does this require some other method than simple algebra?

Just looking at it, L appears to be a shift operator, that is [itex]Lp_t=p_{t-1}[/itex]. Note that [itex]Lp_t[/itex] is not multiplication, but rather the L operator is applied to [itex]t_p[/itex]. Another example of this kind of notation involves the differential operator [itex]D_x[/itex], in particular [itex]D_xf(x)[/itex] is the derivative of f(x) w.r.t. x.